Bosonic String Theory
Bosonic String Theory is the original version of String Theory. It is completely determined by the Polyakov Action, and only includes bosons among it's mass spectrum. It also contains a tachyon in it's mass spectrum as the ground state. Nambu-Goto Action The Nambu-Goto Action is the action principle generally associated with Classical Bosonic String Theory. However, in principle, it is possible to use it for the fully quantised theory as well, but it is much harder. The Nambu-Goto Action can easily be obtained by looking at a Worldsheet suspended in spacetime. It can be stated as: S=-T \int \mbox{d}^2\sigma \sqrt{-\det \gamma_{\alpha\beta} } Where \gamma_{\alpha\beta} is the induced metric on the worldsheet when it is embedded in the background spacetime. This has an obvious geometrical meaning. However, due to the presence of the square root, this action is unsuitable for quantisation. For clarity, observe that under a suitable Diffeomorphism, i.e. reparameterisation, / \ S=-T\int\mbox{d}^2\sigma\sqrt{\left(\dot X X'\right)^2 - \dot X^2 X'^2 } Which is clearly hard to quantise. Polyakov Action The Polyakov Action is an easier-to-quantise action for the Bosonic String Theory. It can be stated as: h^{ab}\partial_aX^\mu\partial_bX^\nu g_{\mu\nu} |cellpadding = 6 |border = 1 |border colour = black |background colour=white}} This Polyakov Action is not so appealing geometrically, but it is easier to quantise. It can be shown that this is classically equivalent to the Nambu-Goto Action as follows (Please click here to view properly): However, as in the Polyakov Action, there is no square root over the X ' s, it is much easier to quantise compared to the Nambu-Goto Action. Virasoro Algebra The L-operator in Bosonic String Theory is given by: }_{m}}=\frac{1}{2}\sum\limits_{n=-\infty }^{\infty } }_{m-n}}\cdot }_{n}}} When m=0 , this results in \hat L_0=\hat H ,. The L-operators form an an algebra, known as the Virasoro Algebra. This is actually a quantisation of the Witt Algebra (also known as the Classical Virasoro Algebra, and is isomorphic to the Conformal Algebra) through a central extension. The multiplication; of this algebra is given, by: \left[ }_{m}}, }_{n}} \right]=\left( m-n \right) }_{m+n}}+\frac{D}{12}m(m-1)(m+1) This algebra is actually useful in removing the negative norm states, as we will see. This algebra also requires the Virasoro Constraints, which are stated simply as: \left. \begin{matrix} }_{0}}\left| \psi \right\rangle =a\left| \psi \right\rangle \\ }_{m>0}}\left| \psi \right\rangle =0 \\ \end{matrix} \right\} Imposing the Virasoro Constrainsts \begin{align} & 0= }_{1}}\left| \Phi \right\rangle \\ & \text{ }= }_{1}} }_{-1}}\left| \right\rangle \\ & \text{ }=\left[ }_{-1}}, }_{1}} \right]\left| \right\rangle + }_{-1}} }_{1}}\left| \right\rangle \\ & \text{ }=\left[ }_{-1}}, }_{1}} \right]\left| \right\rangle \\ & \text{ }=2 }_{0}}\left| \right\rangle \\ & \text{ }=2 \left( a-1 \right)\left| \right\rangle \\ \end{align} Thus, a = 1 /. \begin{align} & \left[ }_{1}}, }_{-2}}+k }_{-1}} }_{-1}} \right]\left| \psi \right\rangle =\left( 3 }_{-1}}+2k }_{0}} }_{-1}}+2k }_{-1}} }_{0}} \right)\left| \psi \right\rangle \text{ } \\ & \text{ }=\left( 3-2k \right) }_{-1}}+4k }_{0}} }_{-1}}\text{ }\left( 3-2k \right) }_{-1}}+4k }_{0}} }_{-1}}\text{ } \\ & 0= }_{1}}\left| \psi \right\rangle = }_{1}}\left( }_{-2}}+k }_{-1}} }_{-1}} \right)\left| \right\rangle =\left( \left( 3-2k \right) }_{-1}}+4k }_{0}} }_{-1}} \right)\left| \right\rangle \\ & \text{ }=\left( \left( 3-2k \right) }_{-1}}+4k }_{-1}}\left( }_{0}}+1 \right) \right)\left| \right\rangle \\ & \text{ }=\left( 3-2k \right) }_{-1}}\left| \right\rangle \\ & 2k=3 \\ & k=\frac{3}{2} \\ \end{align} \begin{align} & }_{2}}\left| \Phi \right\rangle =0 \\ & }_{2}}\left( }_{-2}}+\frac{3}{2} }_{-1}} }_{-1}} \right)\left| \right\rangle =0 \\ & \left[ }_{2}}, }_{-2}}+\frac{3}{2} }_{-1}} }_{-1}} \right]\left| \right\rangle +\left( }_{-2}}+\frac{3}{2} }_{-1}} }_{-1}} \right) }_{2}}\left| \right\rangle =0 \\ & \left[ }_{2}}, }_{-2}}+\frac{3}{2} }_{-1}} }_{-1}} \right]\left| \right\rangle =0 \\ & \left( 13 }_{0}}+9 }_{-1}} }_{-1}}+\frac{D}{2} \right)\left| \right\rangle =0 \\ & \frac{D}{2}=13 \\ & D=26 \\ \end{align} Therefore, D = 26 /. - Mass Spectrum In the Classical Bosonic String Theory, the mass spectrum can be given by m=\sqrt N ; however, in the quantised Bosonic String Theory, it is given by: m=\sqrt{N-a} Since a = 1 , m = \sqrt{N-1} . The first equation is still true in superstring theories, though the second is not, since a is different. Note that this is only for open bosonic strings. For closed bosonic strings, it becomes: m= \sqrt{N+\tilde N-a-\tilde a} =\sqrt{N+\tilde N-2 } p Where we took a = \tilde a = 1 because both the left- (Without the tilde) and right- (with the tilde) moving sectors are from the Bosonic String Theory. Category:String Theory